At the end of First Class, Xavier is paralysed, and him and Magneto split paths. However at the beginning of Last Stand, it flashes back 20 years, and you see the grown up Xavier and Magneto, friends, talking to a young Jean Gray.
Did Xavier get the use of his legs back and then lose it again?
Or is this just a continuity error?
I'd say that this is most likely a continuity error, along the lines of Hank McCoy appearing as a human debating with Sebastian Shaw on TV in "X2," Professor X claiming he built Cerebro with Magneto in "X-Men" when it was in fact built by the government and Beast, Mystique's deep connection with Xavier never being mentioned by Xavier or Magneto in the original trilogy, and Emma Frost being a teenager in "X-Men Origins: Wolverine" when she was an adult in "First Class." I take "First Class" as a reboot, not a prequel.
From the comic books, I remember him getting his legs back at least twice. 
This is from X-Men Deadly Genesis#5
Days of Future Past shows that there is a serum that Xavier and McCoy can use to neutralize their powers, resulting in looking human for McCoy, and walking again for Xavier. But more importantly, the time traveling shenanigans events of DOFP has changed significant parts of the shared timeline of the X-Men movies, including X2 and X3. Notably, Jean and Cyclops are still alive. At this point, we do not know how much of the original trilogy, wolverine origins, or the wolverine have actually occurred as shown.
Well, yes, technnically, what we have here is just a continuity error which, thanks to the timeline having been altered with the intorudction of DoFP, we will never see erased, but with a bit of fanfictioners' trickery, I think it's possible to come up with a fairly plausible answer.
We know that Charles was taking the serum that gave him the ability to walk in 1973. Because he was taking too much of the serum, it stripped him of his psychic abilities. In the DoFP timeline- the altered timeline- Wolverine travels back to the 70's and his involvement effectively ends Charles' addiction to the serum, but in the original timeline, Logan never interacted with Charles in the 70's, so it is concievable that Charles kept on taking the serum well into the 80's. If he lowered his dosage gradually, he would regain hy psychic powers but keep the ability to walk.
As for the fact that he is freind with Erik in the Last Stand flashback; I personally don't think it's very far-fetched that they might have joined forces again temporarily, and then seperated before the events of X-men 1 again, possibly due to their contrary ideals.
It takes place nearly ten years after we see the serum in DOFP so he could have enough time for Charles to make it so Charles can walk and use his powers, but then you could say why doesn’t he walk in xmen well I think his body became immune to it so it stop walking but a a result he remained bald
in x men charles may have been on a low dose of the serume which allowed for a balance between able to walk and telepathy but on a limited power level. or may have used cerebeo to astral project himself to walk with eric. or hank may have desinged a device with helping charles to walk but only works so often. maybe eric was actually using magnetic powers by using a magnet implant in charles legs to aid him. eric repsected charles so much because he never really blamed him for losing his power to walk. i do not think that a immunity to the serum can be developed however i think the dose depends on how stimulated the spinal cord, and muscles would be at the cost of the lesser degree to his powers. hank did indeed imply that if charles did take a minimum dose then he would have powers and walking abillity but may only be half as powerful with telepathy. but charles would still hear voices so he did not want any powers at all. i beleive there may have been a combo of reasons how he can be walking in x men. in infinity universes says infinite methods via science, magic, and assist tools to help him. and as charles got older he just choose to stop walking due to not just wanting to be at full telapath strength but he no longer is a young man and past his physical prime and no one can walk forever so he choose to retire all methods to walk. jean even once offered charles to use her telekinetic powers to restructure Charles spine to allow him to walk. charles said no for several reasons. for one it would be very painful, and two he said it was actually a good reminder that what does not kill you makes you stronger and that to limit himself in some kind of way was poetic and fair and that he no longer wanted to have the need to walk as he tbh has no need. a none cannon timeline. infinite universes, realms, and dimensions so this did happen. in one other timeline he does accept it and gains the abillity to walk however he typically sits in a chair due to weak muscles from not walking, jean uses her powers to make charles muscles grow back stronger, despite all of this charles expresses guilt for taking an easy way out and says he should not be entitled to something that most would actually beg for to have. he then goes on being grateful for being able to walk again. loses his abillity to walk again from a very high speed object. i will call this universe as x men film universe branch reality 135. a nigh all knowing being appears to charles saying that it is his destiny in all realities to lose his abillity to walk
