Her period will end when she is on her way to Makkah but she cannot do ghusl

Firstly: 

If a menstruating woman’s period
ends (and she becomes pure) when she is on the road, and there is still lots
of time left for the prayer, such as if her period ends before Zuhr, or
after it, then she may put Zuhr and ‘Asr together at the time of the later
prayer, and pray them at the time of ‘Asr, and she may delay her ghusl until
that time. Similarly, if her period ends after Maghrib, then she may pray
Maghrib and ‘Isha’ (together) at the time of ‘Isha’, and delay her ghusl
until the time of the prayer. 

But if it is not possible to
delay the prayer, such as if her period ends shortly before Fajr or
afterwards, or before sunset, then in that case she must do ghusl so that
she can pray. 

If she cannot find water, or she
can find it but is afraid to use it, because it is very cold and she has no
means of heating it, or she is afraid that her travelling companions will
move on, it is permissible for her to do tayammum. 

It says in Mataalib Ooli an-Nuha
(1/194), concerning the excuses that make it permissible to do tayammum:

Or he is afraid – if he uses the
water – of some physical harm such as an injury, or because it is very cold
and he cannot find anything with which to heat the water, or he is afraid
that if he uses it his travelling companions will move on. It says in al-Furoo‘:
The apparent meaning is that if he is not afraid of harm resulting from his
travelling companions moving on, because he will miss out on their company.
End quote. 

But in the case of a slight
delay which will not adversely affect the group, then this does not matter
and is not an excuse for not doing ghusl. You have to fear Allah, may He be
exalted, and understand that the basic principle is that you should use
water, and that tayammum is only permissible when there is an excuse, so you
should not be too careless about that. 

In the event that the concession
of tayammum is valid in your case, then if you do tayammum with the
intention of purifying yourself from both menses and minor impurity, you do
not need to do wudoo’ after tayammum. But if you do tayammum to purify
yourself from menses only, then you must do wudoo’ after that, and that is
preferable, in sha Allah, because water is available and you will not be
harmed by doing wudoo’ with it. 

Shaykh ‘Abd al-‘Azeez ibn Baaz
(may Allah have mercy on him) was asked: 

If a person falls asleep in Mina
and needs water for ghusl, but because of the high numbers of pilgrims there
is little water, what should he do instead of using water? What if he wants
to pray when he is junub? Is it acceptable for him to purify himself with
dust? I hope that you can explain this matter, because many people are faced
with such issues. 

He (may Allah have mercy on him)
replied: 

If a person has a wet dream in
Muzdalifah or any place where there are a lot of people, he should look for
water, even if he has to pay for it, from any place he can, on the edges of
Mina in any direction, or if he finds a place where he can do ghusl for
free, then he must do that. 

But if that is not possible,
whether in return for payment or for free, and he cannot find any place to
do ghusl, then he must do tayammum with dust and pray. He should strike the
dust with his hands and wipe his face and hands with the intention of
purifying himself from janaabah and from minor impurity. In other words, he
should intend the tayammum to purify himself from both, the minor impurity
which necessitated wudoo’ and the major impurity which necessitated ghusl
from janaabah. He should intend both, and that will suffice, then he may
pray, and his prayer is valid, praise be to Allah. Fear Allah as much as you
can, for Allah says (interpretation of the meaning): “and you find no
water, perform Tayammum with clean earth and rub therewith your faces and
hands (Tayammum)” [an- Nisa’
4:43].
But he has to try hard and not be careless; he has to try hard and ask. If
he finds water, even if he has to pay for it and collect it in a bucket and
the like, he should do that, then move away from people, and look for a
tent, if there is a tent in which there is no one, or a place where no one
can see him, and do ghusl.

End quote from Fataawa Noor
‘ala ad-Darb by Shaykh Ibn Baaz (may Allah have mercy on him), (5/276).
 

Secondly: 

With regard to your question
about tayammum and whether it is to be done for every prayer or, if it is
done to remove minor impurity once, that is sufficient. The scholars
differed concerning that as there are two views: 

The first view is that it is not
required for every prayer, because tayammum – like wudoo’ – removes
impurity, so it is not invalidated by the ending of the time for the prayer;
rather it is invalidated by the things that are well-known to invalidate
wudoo’. 

This is the view of the Hanafis
and is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have
mercy on him). It was regarded as more correct by Shaykh Ibn ‘Uthaymeen (may
Allah have mercy on him) – ash-Sharh al-Mumti‘ (1/314) 

They quoted several things as
evidence, including the verse in which Allah, may He be exalted, says
(interpretation of the meaning):

“If you are in a state of
Janaba (i.e. had a sexual discharge), purify yourself (bathe your whole
body). But if you are ill or on a journey or any of you comes from answering
the call of nature, or you have been in contact with women (i.e. sexual
intercourse) and you find no water, then perform Tayammum with clean earth
and rub therewith your faces and hands. Allah does not want to place you in
difficulty, but He wants to purify you”

[al-Maa’idah 5:6]. 

And the Prophet (blessings and
peace of Allah be upon him) said: “The earth has been made a place of
prostration and a means of purification for me.” (Agreed upon). Tayammum has
been ordained as a means of purification, so how can it be said that it does
not purify or that it does not remove impurity? Rather it is only a means to
make it permissible to pray. 

The second scholarly view is: 

That tayammum makes it
permissible to pray and so on, but it does not remove impurity. This is the
view of the majority of Maalikis, Shaafa‘is and Hanbalis. Based on that,
tayammum should be done after the time for the prayer begins, and it becomes
invalid when the time for the prayer ends. So if a person does tayammum for
Zuhr prayer, he may offer that prayer and whatever he wants of naafil
prayers, but when the time for Zuhr ends, he must do tayammum again for ‘Asr
prayer. 

Undoubtedly it is preferable to
be on the safe side with regard to such matters, especially when there is no
hardship involved in doing tayammum for every prayer. 

When you reach Makkah, it is
permissible for you to pray Taraweeh first, and to delay ‘umrah until a time
when the crowds are less, or to delay it so that you can rest and regain
strength and energy, so long as you avoid the things that are prohibited
when in ihram. 

And Allah knows best.